There are, in fact, plenty of plugins that provide a mailing list functionality that work just fine with WordPress. Three things come to mind here:

1) Be patient. Depending on your server and host, it could take several hours for emails to be sent and received. (If you have a large mailing list, your host may only allow so many emails to be sent per hour, to prevent overloading their mail servers.)

2) Your host may be blocking you from sending outgoing mail in quantity, since they don't want to be perceived as sending out spam.

3) Switch from the default mail port of 25 to something else. SMTP uses port 25 by default, and this is sometimes blocked by default on some servers, again to prevent spam. This article may help you understand this. Alternately, it may also confuse you. The important point is to find out if your host is blocking port 25, and if they are, how to switch it to something else.